Effects of impulsive harvesting and an evolving domain in a diffusive logistic model
EEffects of impulsive harvesting and an evolvingdomain in a diffusive logistic model ∗ Yue Meng , Zhigui Lin † , and Michael Pedersen Abstract.
In order to understand how the combination of domain evolutionand impulsive harvesting affect the dynamics of a population, we propose adiffusive logistic population model with impulsive harvesting on a periodicallyevolving domain. Initially the ecological reproduction index of the impulsiveproblem is introduced and given by an explicit formula, which depends onthe domain evolution rate and the impulsive function. Then the thresholddynamics of the population under monotone or nonmonotone impulsive har-vesting is established based on this index. Finally numerical simulations arecarried out to illustrate our theoretical results, and reveal that a large domainevolution rate can improve the population survival, no matter which impul-sive harvesting takes place. Contrary, impulsive harvesting has a negativeeffect on the population survival, and can even lead to the extinction of thepopulation.
MSC:
Keywords:
Impulsive harvesting; Evolving domain; Ecological reproductionindex; Threshold dynamics; Persistence and extinction
Mathematical models, in which reaction diffusion models often focus on the spread andpersistence of a population [4], have been widely used to study ecological phenomena[26]. In nature, species grow and move randomly like e.g. migration of animals, expan-sion of invaders and so on. Fish and large mammals give birth in a specific period, so thepopulation experience a birth impulsive growth [5]. On the other hand, in order to pro-mote a sustainable development of ecosystems or obtain sustainable economic benefits, ∗ The work is partially supported by the NNSF of China (Grant No. 11771381, 61877052). † Corresponding author. Email: [email protected] (Z. Lin). a r X i v : . [ m a t h . A P ] D ec opulation systems are often disturbed by human management and development strate-gies [7], such as planting and harvesting, vaccination or the release of natural enemies.Such stages of disturbance typically occur for a short period of time, but have significantimpacts on the population density and numbers. Classical differential equations are notwell suited to describe such phenomena, where important drivers are non-continuousprocesses. Hence impulsive differential equations have been introduced to model andcharacterize these hybrid discrete-continuous processes.The theoretical research of impulsive differential equations began by work of Mil’manand Myshkis in the 1960s [25], and has further developed since the 80s. An impulsivedifferential system is a special dynamical system, which has characteristics of both con-tinuous systems and discrete systems. Its remarkable characteristic is that it can induceshort-term rapid changes on the state of systems, depending on time and state. Im-pulsive ordinary differential equations have been widely studied in population ecologicaldynamics, such as predator-prey systems [33], pest management systems [34], systemswith control strategies [15] and so on. Since drugs are often introduced into the body inthe form of pulses by oral or injection in the treatment of diseases, impulsive ordinarydifferential equations have also been used to analyze the dynamics of infectious diseases,see [9, 14, 32] and references therein.Recently, due to the complex dynamics generated by pulses, it has become an inter-esting focus of attention how pulses affect a reaction diffusion system. Some scholarshave studied the effects of various responses on the dynamics of impulsive reaction dif-fusion predator-prey systems [22] and the impacts of seasonal variation on their patternformation [37]. Lewis and Li [17] investigated a reaction diffusion model with a seasonalbirth pulse and explored how the birth pulse affects the dynamics of the population,including spreading speed, minimal domain size, travelling waves and complex bifurca-tions. Considering a nonlocal dispersal stage based on the work in [17], Wu and Zhao[38] established threshold-type dynamics of the system when the domain is bounded andproved the existence of a spreading speed in an unbounded domain. Liang, et al. [19]extended the growth population model of pest with multiple pulses, including birth andpesticide applications. They analyzed effects of the birth rate and the killing efficacyon a traveling wave and spreading speed, and also investigated the optimal timing ofuse of pesticides, pest control strategy and spatial dependent killing rates by numericalsimulations.Previous studies on impulsive differential equations have been carried out on fixeddomains. Space is identified as a functional property of ecological processes in nature[35], and it does play an important role in the dispersal of the population. For example,habitat structure can enhance the persistence of the species [11]. It has been well doc-umented that a spatially inhomogeneous environment affects the population dynamics,disease transmission and species evolution [23]. With the increased focus on the role ofhabitats in ecology, the influence of shifting domains on species dynamics has becomea concern recently, due to the obvious fact that habitats are not constant in nature.Free boundary problems, modeling situations where shifting boundaries of the habitatare unknown and affected by the movement of the population, are used to describe in-vasive species [10, 18] and the spread of infectious diseases [21]. On the other hand, it is2ell-known that habitats where species live on, often present cyclic variation, which isinfluenced by environment factors such as rainfall, temperature and so on. For instance,the increase of rainfall can reduce the riparian habitat area of waterbirds, in cases whererainfall shows a long-term cyclic behavior, like on the Pampas region of Argentina [3].Such domains, where the changing boundaries are known, are called periodically evolvingdomains.Motivated by the above, in order to explore how an evolving domain and impulsiveharvesting affect the dynamics of a population, we consider a diffusive logistic modelwith impulsive harvesting on a periodically evolving domain. This model describes thesituation where a harvesting pulse, which is described by a function g , takes place atevery time nT ( n = 0 , , , ... ) during the continuous growth and dispersal process of apopulation. The density of the population at the end of the harvesting stage is givenby the function g applied to the density of the population at the beginning of the stage.Since 0 < g ( u ) /u < − g ( u ) /u denotes harvesting rate. During thedispersal stage, the population diffuses by the coefficient d ( >
0) and follows a logisticequation. α ( >
0) represents the intrinsic growth rate of the population and the effect ofthe interspecific competition among individuals is denoted by γ ( > u ( t, x ) be thedensity of the population at time t and point x ∈ [0 , l ] with initial density u (0 , x ) = u ( x ).With a Dirichlet boundary condition, the population model in a one-dimensional spaceis introduced as follows: ∂u∂t = d ∂ u∂x + αu − γu , x ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,u ( t,
0) = u ( t, l ) = 0 , t > ,u (0 , x ) = u ( x ) ≥ , (cid:54)≡ , x ∈ [0 , l ] ,u (( nT ) + , x ) = g ( u ( nT, x )) , x ∈ (0 , l ) , n = 0 , , , ... (1.1)In this paper, we make the following assumption about the pulse function g :(A1) g ( u ) is a once continuously differentiable function for u ≥ , g (0) = 0 , g (cid:48) (0) > , and for u > , g ( u ) > , g ( u ) /u is nonincreasing with respect to u and < g ( u ) /u < . The pulse functions g satisfying (A1) usually takes the form ( including the Beverton-Holt function): g ( u ) = mua + u (1.2)with m > a > g ( u ) = ue r − bu (1.3)with r > b > , l ( t )) be a periodically evolving domain at time t > l ( t ) . u ( t, x ( t )) denotes the population density at time t > x ( t ) ∈ (0 , l ( t )).Let x ( t ) = m ( t ) and x ( t ) = n ( t )( m ( t ) ≤ n ( t )) be two arbitrary end points of theinterval where x ( t ) varies from 0 to l ( t ). Then, ( m ( t ) , n ( t )) ⊂ (0 , l ( t )) denotes an evolvingdomain at time t > m ( t ) and ending point n ( t ). By theprinciple of mass conservation we have that ddt (cid:82) n ( t ) m ( t ) u ( t, x ( t )) dx = du x ( t, n ( t )) − du x ( t, m ( t )) + (cid:82) n ( t ) m ( t ) f ( t, u ( x ( t ))) dx, = (cid:82) n ( t ) m ( t ) ( du xx + f ( u )) dx, (1.4)where f ( u ) = αu − γu . We further assume that the evolution of the domain is uniformand isotropic, in other words, the domain evolves by the same ratio in all directions astime increases. One possibility can be described as x ( t ) = ρ ( t ) y, y ≥ , (1.5)where the positive continuous function ρ ( t ) is called evolution rate. Here, because of theperiodic evolution of the domain (0 , l ( t )), ρ ( t ) is T-periodic in time i.e. ρ ( t + T ) = ρ ( t ) (1.6)for some T > ρ (0) = 1. A study of periodically evolving domains can be found ine.g. [16, 28]. If ˙ ρ ( t ) ≥
0, the domain is called a growing domain, see [24] and referencestherein, and a shrinking domain ( ˙ ρ ( t ) ≤
0) has been discussed in reference [36] andothers.Let l (0) = l , then the evolving domain can be written (0 , l ( t )) = (0 , ρ ( t ) l ). Set m ( t ) = ρ ( t ) y and n ( t ) = ρ ( t ) y , where y , y ∈ (0 , l ). Noting that u ( t, x ( t )) = u ( t, ρ ( t ) y ), we define u ( t, x ( t )) = v ( t, y ), which together with (1.5) yields u x = 1 ρ ( t ) v y , u xx = 1 ρ ( t ) v yy . (1.7)From the left hand side of (1.4), we have ddt (cid:82) n ( t ) m ( t ) u ( t, x ( t )) dx = ddt (cid:82) ρ ( t ) y ρ ( t ) y u ( t, ρ ( t ) y ) d ( ρ ( t ) y ) , = ddt (cid:82) y y v ( t, y ) ρ ( t ) dy, = (cid:82) y y ( v t ( t, y ) ρ ( t ) + v ( t, y ) ˙ ρ ( t )) dy. (1.8)Using (1.7), the right hand side becomes (cid:82) n ( t ) m ( t ) ( du xx + f ( u )) dx = (cid:82) y y ( dρ ( t ) v yy + f ( v ( t, y ))) ρ ( t ) dy. (1.9)4rom (1.8) and (1.9), equation (1.4) can be written as follows (cid:90) y y ( v t + ˙ ρ ( t ) ρ ( t ) v ) dy = (cid:90) y y ( dρ ( t ) v yy + f ( v ( t, y ))) dy, t > . (1.10)Since m ( t ) and n ( t ) are arbitrary, then (1.10) holds for any y , y ∈ (0 , l ). Therefore,(1.1) is transformed into the following problem on a fixed domain v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v − γv , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = v ( y ) ≥ , (cid:54)≡ , y ∈ (0 , l ) ,v (( nT ) + , y ) = g ( v ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ... (1.11)When g ( v ) = v , that is, the case when impulsive harvesting does not occur, (1.11)is reduced to a classical logistic equation on a periodically evolving domain, previouslydiscussed by Jiang and Wang [16]. They proved persistence and extinction of speciesbased on the critical value D := aTλ (cid:82) T ρ − ( t ) dt , where λ ( >
0) is the principal eigenvalueof − ∆ in (0 , l ) under the Dirichlet boundary condition. The results indicate that speciesbecome extinct when d ∈ ( D , + ∞ ), on the other hand, species persist if d ∈ (0 , D ).They also analyzed effects of the evolution rate on the persistence of species.We are interested in what amusing changes the impulsive harvesting will impose onthe system (1.11), whether a new threshold value like D in [16] in terms of the pulsefunction can be introduced to establish threshold-type results for the persistence andextinction of the population, and whether these results are consistent with results in [16].This paper also aims to explore how regional evolution affects the dynamic behaviour ofthe population when impulsive harvesting takes place, and is organized as follows: In thenext section, a new threshold value is introduced, which is the ecological reproductionindex of the problem with impulses. Moreover, we provide an explicit formula of theecological reproduction index and analyze the relationship between domain evolutionand this index. Section 3 is concerned with threshold-type results on the asymptoticbehaviours of the solution to problem (1.11) when the impulsive harvesting is eithermonotone or not. Numerical simulations are carried out to understand the effects of theevolution rate and impulsive harvesting on the dynamics of the population, and thenbiological explanations are given in the Section 4. The last section contains a discussionsof the results. In this section, a new threshold value, which is the ecological reproduction index of theproblem with impulses, is introduced and analyzed.Linearizing problem (1.11) at v = 0, which represents the disease-free equilibrium5pidemiologically, we obtain the following linearized system v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v ( t,
0) = v ( t, l ) = 0 , t > ,v (( nT ) + , y ) = g (cid:48) (0) v ( nT, y ) , y ∈ (0 , l ) , n = 0 , , , ... (2.1)We first consider the auxiliary system as follows (cid:40) v t = ( α − ˙ ρ ( t ) ρ ( t ) ) v, t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v (( nT ) + ) = g (cid:48) (0) v ( nT ) , n = 0 , , , ... (2.2)As in [1], let E ( t, s ) be the evolution operator of the problem (cid:40) v t = − ˙ ρ ( t ) ρ ( t ) v, t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v (( nT ) + ) = g (cid:48) (0) v ( nT ) , n = 0 , , , ... (2.3)Then, the evolution operator E ( t, s ) can be written as E ( t, s ) = e − (cid:82) ts ˙ ρ ( τ ) ρ ( τ ) dτ ( g (cid:48) (0)) k , where k represents the number of impulsive points on [ s, t ). Since (cid:82) ts ˙ ρ ( τ ) ρ ( τ ) dτ is boundedfor any t > s , there exists a positive constant K such that (cid:107) E ( t, s ) (cid:107)≤ K, t ≥ s, s ∈ R . Let C T be the Banach space given by C T = (cid:8) ω | ω ∈ C (( nT, ( n + 1) T ]) , ω ( t + T ) = ω ( t ) for t ∈ R , ω (( nT ) + ) = ω ((( n + 1) T ) + ) , n ∈ Z (cid:9) with the norm (cid:107) ω (cid:107) = sup t ∈ [0 ,T ] | ω ( t ) | and the positive cone C + T := { ω ∈ C T | ω ( t ) ≥ , ∀ t ∈ R } . A linear operator on C T can be introduced by[ Lω ]( t ) = (cid:90) ∞ αE ( t, t − s ) ω ( t − s ) ds. It is clear that L is positive and compact on C T . According to [1], we define the spectralradius of L (cid:60) = r ( L )as the basic reproduction number of the periodic impulsive system (2.2). Moreover,Theorem 2 in [1] shows that (cid:60) = µ , where µ satisfies the following problem v t = ( αµ − ˙ ρ ( t ) ρ ( t ) ) v, t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v (( nT ) + ) = g (cid:48) (0) v ( nT ) , n = 0 , , , ...,v (0) = v ( T ) . φ t = dρ ( t ) φ yy + ( αR − ˙ ρ ( t ) ρ ( t ) ) φ, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,φ ( t,
0) = φ ( t, l ) = 0 , t > ,φ (0 , y ) = φ ( T, y ) , y ∈ [0 , l ] ,φ (( nT ) + , y ) = g (cid:48) (0) φ ( nT, y ) , y ∈ (0 , l ) , n = 0 , , , ... (2.4)The theory for the basic reproduction number for the impulsive reaction-diffusion systemis not fully established, but similar to the above, we introduce the ecological reproductionindex R for our impulsive system by solving problem (2.4), which can give the explicitformula of R and find a corresponding positive eigenfunction φ ( t, y ). Theorem 2.1
The ecological reproduction index of problem (1 . can be explicitly ex-pressed as R = α dλ T (cid:82) T ρ t ) dt − T ln g (cid:48) (0) , (2.5) where λ ( > is the principal eigenvalue of − ∆ in (0 , l ) under Dirichlet boundarycondition, and R is monotonically increasing with respect to ρ ( t ) . Proof:
Let φ ( y, t ) = f ( t ) ψ ( y ) , where ψ ( y ) is the eigenfunction related to λ in the eigenvalue problem (cid:40) − ψ (cid:48)(cid:48) = λ ψ, y ∈ (0 , l ) ,ψ (0) = ψ ( l ) = 0 , (2.6)then problem (2.4) becomes f (cid:48) ( t ) ψ ( y ) = dρ ( t ) f ( t ) ψ yy ( y ) + ( αR − ˙ ρ ( t ) ρ ( t ) ) f ( t ) ψ ( y ) , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,ψ (0) = ψ ( l ) = 0 ,f (0) = f ( T ) ,f (( nT ) + ) = g (cid:48) (0) f ( nT ) , n = 0 , , , ... (2.7)By separating variables, it follows from the first equation of problem (2.7) that f (cid:48) ( t ) + ( ˙ ρ ( t ) ρ ( t ) − αR ) f ( t ) dρ ( t ) f ( t ) = ψ (cid:48)(cid:48) ( y ) ψ ( y ) = − λ . Then, the first equation in problem (2.7) becomes f (cid:48) ( t ) + ( ˙ ρ ( t ) ρ ( t ) − αR + dλ ρ ( t ) ) f ( t ) = 0 . (2.8)By solving equation (2.8), we find f ( t ) = Ce (cid:82) t − ˙ ρ ( τ ) ρ ( τ ) + αR − dλ ρ τ ) dτ , t ∈ (0 + , T ] , here the initial value C satisfies C = f (0 + ) = g (cid:48) (0) f (0). Then, f ( T ) = g (cid:48) (0) f (0) e (cid:82) T − ˙ ρ ( τ ) ρ ( τ ) + αR − dλ ρ τ ) dτ , which together with (1.6) and the third equation in problem (2.7) yields g (cid:48) (0) e αTR − dλ (cid:82) T ρ t ) dt = 1 . Therefore, R = α dλ T (cid:82) T ρ ( t ) dt − T ln g (cid:48) (0) . From the explicit formula (2.5), the monotonicity is easily obtained.We note that when g ( v ) = v , then R becomes R := αTdλ (cid:82) T ρ ( t ) dt , (2.9)which is consistent with the threshold value in problem without impulses discussed in [16]. Remark 2.1
When g (cid:48) (0) ≤ , it is obvious that the ecological reproduction index R > . If g (cid:48) (0) > , the positivity of R can not be guaranteed, and then R as an ecological reproductionindex makes no sense. In this situation we can consider the following problem: v t = dρ ( t ) v yy + ( α + M − ˙ ρ ( t ) ρ ( t ) ) v − γv − M v, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = v ( y ) ≥ , (cid:54)≡ , y ∈ (0 , l ) ,v (( nT ) + , y ) = g ( v ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ..., (2.10) which is equivalent to problem (1.11) . Then, the corresponding periodic eigenvalue problem isas follows φ t = dρ ( t ) φ yy + ( α + MR ∗ − ˙ ρ ( t ) ρ ( t ) ) φ − M φ, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,φ ( t,
0) = φ ( t, l ) = 0 , t > ,φ (0 , y ) = φ ( T, y ) , y ∈ [0 , l ] ,φ (( nT ) + , y ) = g (cid:48) (0) φ ( nT, y ) , y ∈ (0 , l ) , n = 0 , , , ... (2.11) Therefore, the ecological reproduction index R ∗ := α + M dλ T (cid:82) T ρ t ) dt − T ln g (cid:48) (0)+ M , where M = T | ln g (cid:48) (0) | can be chosen to make sure that R ∗ > . The threshold-type dynamics of problem (1.11) , whichis established based on the new ecological reproduction index R ∗ , is the same as if based on R . The following result holds for reaction diffusion problems without impulses [20, 39], andalso holds for our impulsive problem:
Lemma 2.2 sign( R −
1) = sign λ ∗ , where λ ∗ satisfies φ t = dρ ( t ) φ yy + ( α − ˙ ρ ( t ) ρ ( t ) ) φ − λ ∗ φ, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,φ ( t,
0) = φ ( t, l ) = 0 , t > ,φ (( nT ) + , y ) = g (cid:48) (0) φ ( nT, y ) , y ∈ (0 , l ) , n = 0 , , , ... In fact, λ ∗ = T ln g (cid:48) (0) + α − dλ T (cid:82) T ρ ( t ) dt . The result is obvious. Asymptotic behaviors of the solution
In this section, we first explore the asymptotic behaviors of the solution to problem (1.11)based on the threshold value R in the case where g is monotone, and then we consider thenonmonotone case. We start by stating the assumptions:(B1) g ( u ) is nondecreasing for u ≥ . (B2) There are positive constants D , σ < θ , and ν > such that g ( u ) ≥ g (cid:48) (0) u − Du ν for ≤ u ≤ σ . It is obvious that the Beverton-Holt function (1.2) satisfies the assumptions (B1) and (B2).We claim here that problem (1.11) admits a global classical solution v ( t, y ), that is, v ( t, y )is once continuously differentiable in t ( t ∈ (0 , + ∞ )), and twice continuously differentiable in y ( y ∈ (0 , l )). In fact, the initial value v ( y ) ∈ C ([0 , l ]) and the fact that g is once continuouslydifferentiable implies that v (0 + , y ) ∈ C ([0 , l ]). By virtue of standard theory for parabolicequations, we can deduce that v ( t, y ) ∈ C , ((0 , T ] × (0 , l )). Then, v ( T + , y ) = g ( v ( T, y ))is also once continuously differentiable in y . Hence, let v ( T + , y ) be a new initial value for t ∈ ( T + , T ], then v ( t, y ) ∈ C , (( T, T ] × (0 , l )). Eventually, we can obtain the solution v ( t, y ) of problem (1.11) for t ≥ y ∈ (0 , l ) by the same procedures.Then we define P C ([0 , + ∞ ) × [0 , l ]) = { v ( t, y ) | v ( t, y ) ∈ C (( nT, ( n + 1) T ] × [0 , l ]) } ,P C , ((0 , + ∞ ) × (0 , l )) = { v ( t, y ) | v ( t, y ) ∈ C , (( nT, ( n + 1) T ] × (0 , l )) } . The definition of upper and lower solutions and the comparison principle for the initial bound-ary problem (1.11) with impulses are given as follows:
Definition 3.1
We say that ˜ v ( t, y ) , ˆ v ( t, y ) ∈ P C , ((0 , + ∞ ) × (0 , l )) ∩ P C ([0 , + ∞ ) × [0 , l ]) satisfying ≤ ˆ v ( t, y ) ≤ ˜ v ( t, y ) are upper and lower solutions of problem (1 . , respectively, if ˜ v ( t, y ) and ˆ v ( t, y ) make the following relationship true: ˜ v t ≥ dρ ( t ) ˜ v yy + ( α − ˙ ρ ( t ) ρ ( t ) )˜ v − γ ˜ v , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , ˆ v t ≤ dρ ( t ) ˆ v yy + ( α − ˙ ρ ( t ) ρ ( t ) )˜ v − γ ˆ v , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , ˆ v ( t,
0) = 0 ≤ ˜ v ( t, , ˆ v ( t, l ) = 0 ≤ ˜ v ( t, l ) , t > , ˜ v (( nT ) + , y ) ≥ g (˜ v ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ... ˆ v (( nT ) + , y ) ≤ g (ˆ v ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ... (3.1) with the initial condition ≤ ˆ v (0 , y ) ≤ v ( y ) ≤ ˜ v (0 , y ) , y ∈ [0 , l ] . (3.2) Lemma 3.1 (Comparison principle) Let ˜ v ( t, y ) and ˆ v ( t, y ) be the upper and lower solutions toproblem (1.11) , then any solution v ( t, y ) of problem (1 . satisfies ˆ v ( t, y ) ≤ v ( t, y ) ≤ ˜ v ( t, y ) , t ∈ [0 , ∞ ) , y ∈ [0 , l ] . roof: Let w ( t, y ) = ˜ v ( t, y ) − v ( t, y ). By the definition of an upper solution to problem (1.11)and assumption (B1), we have that: w ( t, y ) ≥ dρ ( t ) w yy ( t, y ) + ( α − ˙ ρ ( t ) ρ ( t ) ) w ( t, y ) − γ ( w ( t, y )) , y ∈ (0 , l ) , t ∈ (0 + , T ] ,w ( t, ≥ , w ( t, l ) ≥ , t > ,w (0 + , y ) = ˜ v (0 + , y ) − v (0 + , y ) ≥ g (˜ v (0 , y )) − g ( v ( y )) ≥ , y ∈ (0 , l ) . The maximum principle implies that w ( t, y ) ≥ t ∈ (0 + , T ] and y ∈ [0 , l ]. Then,˜ v ( t, y ) ≥ v ( t, y ) holds for t ∈ (0 + , T ] and y ∈ [0 , l ]. We take w ( t, y ) at time t = T + as anew initial value for the period ( T + , T ], which satisfies w ( T + , y ) = ˜ v ( T + , y ) − v ( T + , y ) ≥ g (˜ v ( T, y )) − g ( v ( T, y )) ≥
0. By the same procedures, ˜ v ( t, y ) ≥ v ( t, y ) can also be obtained for t ∈ ( T + , T ] and y ∈ [0 , l ]. Step by step, ˜ v ( t, y ) ≥ v ( t, y ) holds for all t ≥ y ∈ [0 , l ].Similarly, the result that v ( t, y ) ≥ ˆ v ( t, y ) for all t ≥ y ∈ [0 , l ] can also be deduced.The abovementioned preliminaries allow us to investigate the asymptotic behaviours of thesolution to problem (1.11). Theorem 3.2 If R < , then the solution v ( t, y ) of problem (1 . satisfies lim t →∞ v ( t, y ) = 0 uniformly for y ∈ [0 , l ] . Proof:
Let v ( t, y ) = Ke α (1 − R ) t φ ( t, y ) , where φ ( t, y )( ≤
1) is a positive normalized eigenfunction of problem (2.4) and K is a positiveconstant to be chosen later. It is easy to verify that v ( t, y ) is a solution of the following linearproblem v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v, y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = Kφ (0 , y ) , y ∈ (0 , l ) ,v (( nT ) + , y ) = g (cid:48) (0) v ( nT, y ) , y ∈ (0 , l ) , n = 0 , , , ... (3.3)It follows from (A1) that g ( v ) v ≤ lim ε → g ( ε ) − g (0) ε = g (cid:48) (0)holds for sufficiently small ε >
0, which implies that v (( nT ) + , y ) = g (cid:48) (0) v ( nT, y ) ≥ g ( v ( nT, y )) . (3.4)Meanwhile, for any initial function v (0 , y ), a sufficiently large constant K can be chosen suchthat v (0 , y ) ≥ v (0 , y ). Since the reaction term in problem (3.3) is larger than that in problem(1.11), we conclude that v ( t, y ) is a supersolution of problem (1.11). It then follows fromLemma 3.1 that v ( t, y ) ≤ v ( t, y ) , t ≥ , y ∈ [0 , l ] . If R <
1, then lim t →∞ v ( t, y ) = 0 for all y ∈ [0 , l ]. Therefore, lim t →∞ v ( t, y ) = 0 uniformly for y ∈ [0 , l ]. ext, in order to study the asymptotic behaviors of the solution to problem (1.11) when R >
1, we study the following auxiliary periodic problem and state the relationship betweenits periodic solution and the solution of problem (1.11). v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v − γv , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = v ( T, y ) , y ∈ (0 , l ) ,v (( nT ) + , y ) = g ( v ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ... (3.5)The existence of a solution to the periodic problem (3.5) is given by upper and lower solutionstechnique. Definition 3.2
We call ˜ v ( t, y ) , ˆ v ( t, y ) ∈ P C , ((0 , + ∞ ) × (0 , l )) ∩ P C ([0 , + ∞ ) × [0 , l ]) sat-isfying ≤ ˆ v ( t, y ) ≤ ˜ v ( t, y ) an upper and lower solution of problem (3 . , respectively, if ˜ v ( t, y ) and ˆ v ( t, y ) satisfy relationship (3 . and periodic conditions ˆ v (0 , y ) ≤ ˆ v ( T, y ) , ˜ v (0 , y ) ≥ ˜ v ( T, y ) , y ∈ [0 , l ] . (3.6)Obviously the upper and lower solutions of the periodic problem (3.5) are also the upper andlower solutions of the initial value problem (1.11) provided that ˆ v ( t, y ) ≤ v (0 , y ) ≤ ˜ v ( t, y ) , y ∈ [0 , l ].Let f ( v, t ) = ( α − ˙ ρ ( t ) ρ ( t ) ) v − γv and choose K ∗ = αγ + sup t ∈ [0 ,T ] | ˙ ρ ( t ) | ρ ( t )such that F ( v, t ) = K ∗ v + f ( v, t ) is monotonically nondecreasing with respect to v . If thereexists upper and lower solutions ˜ v and ˆ v of problem (3.5), using v (0) = ˜ v and v (0) = ˆ v as initialiteration, we can construct the iteration sequences { v ( m ) } and { v ( m ) } by the following process v ( m ) t − dρ ( t ) v ( m ) yy + K ∗ v ( m ) = + K ∗ v ( m − +( α − ˙ ρ ( t ) ρ ( t ) ) v ( m − − γ ( v ( m − ) , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,v ( m ) t − dρ ( t ) v ( m ) yy + K ∗ v ( m ) = K ∗ v ( m − +( α − ˙ ρ ( t ) ρ ( t ) ) v ( m − − γ ( v ( m − ) , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] ,v ( m ) ( t,
0) = v ( m ) ( t, l ) = v ( m ) ( t,
0) = v ( m ) ( t, l ) = 0 , t > ,v ( m ) (0 , y ) = v ( m − ( T, y ) , v ( m ) (0 , y ) = v ( m − ( T, y ) , y ∈ (0 , l ) ,v ( m ) (( nT ) + , y ) = g ( v ( m − (( n + 1) T, y )) , y ∈ (0 , l ) , n = 0 , , , ...,v ( m ) (( nT ) + , y ) = g ( v ( m − (( n + 1) T, y )) , y ∈ (0 , l ) , n = 0 , , , ... (3.7)Similarly as in [27], the monotone property of the iteration sequences is obtained in thefollowing lemma. Lemma 3.3
If there exists upper and lower solutions ˜ v and ˆ v of problem (3 . , then the se-quences { v ( m ) } and { v ( m ) } have the monotone property ˆ v ≤ v ( m ) ≤ v ( m +1) ≤ v ( m +1) ≤ v ( m ) ≤ ˜ v for t ∈ [0 , + ∞ ) and y ∈ [0 , l ] . roof: Let w (0) = v (1) − v (0) = v (1) − ˆ v . By (3.1), (3.6) and (3.7), we have w (0) (0 , y ) = v (1) (0 , y ) − v (0) (0 , y ) = v (0) ( T, y ) − ˆ v (0 , y ) ≥ , y ∈ [0 , l ] (3.8)and w (0) − dρ ( t ) w (0) yy + K ∗ w (0) = K ∗ v (0) + f ( v (0) , t ) − (ˆ v t − dρ ( t ) ˆ v yy + K ∗ ˆ v )= f (ˆ v, t ) − (ˆ v t − dρ ( t ) ˆ v yy ) ≥ , y ∈ (0 , l ) , t ∈ (0 + , T ] ,w (0) ( t,
0) = v (1) ( t, − ˆ v ( t,
0) = 0 , w (0) ( t, l ) = v (1) ( t, l ) − ˆ v ( t, l ) = 0 , t > ,w (0) (0 + , y ) = v (1) (0 + , y ) − ˆ v (0 + , y ) = g ( v (0) ( T, y )) − g (ˆ v (0 , y )) ≥ , y ∈ (0 , l ) . (3.9)(3.9) and the positivity lemma for parabolic problems show that w (0) ≥ t ∈ (0 , T ] and y ∈ [0 , l ], which together with (3.8) yields that w (0) ≥ t ∈ [0 , T ] and y ∈ [0 , l ] i.e. v (1) ≥ v (0) for t ∈ [0 , T ] and y ∈ [0 , l ]. Then, if w (0) ( T + , y ) is a new initial value for t ∈ ( T, T ],we can deduce that v (1) ≥ v (0) for t ∈ ( T, T ] and y ∈ [0 , l ] similarly. Hence v (1) ≥ v (0) holdsfor t ∈ [0 , + ∞ ) and y ∈ [0 , l ]. Similarly, the property of an upper solution gives the resultthat v (1) ≥ v (0) for t ∈ [0 , + ∞ ) and y ∈ [0 , l ]. Moreover, let w (1) = v (1) − v (1) , then w (1) − dρ ( t ) w (1) yy + K ∗ w (1) = F ( v (0) , t ) − F ( v (0) , t ) ≥ , y ∈ (0 , l ) , t ∈ (0 + , T ] ,w (1) ( t,
0) = v (1) ( t, − v (1) ( t,
0) = 0 , w (1) ( t, l ) = v (1) ( t, l ) − v (1) ( t, l ) = 0 , t > ,w (1) (0 , y ) = v (1) (0 , y ) − v (1) (0 , y ) = ˜ v ( T, y ) − ˆ v ( T, y ) ≥ , y ∈ (0 , l ) ,w (1) (0 + , y ) = v (1) (0 + , y ) − v (1) (0 + , y ) = g ( v (0) ( T, y )) − g ( v (0) ( T, y )) ≥ , y ∈ (0 , l ) , which yields that w (1) ≥ t ∈ [0 , T ] and y ∈ [0 , l ]. The result that w (1) ≥ t ∈ [0 , + ∞ )and y ∈ [0 , l ] can be deduced by the same procedures. Therefore, we obtain that v (0) ≤ v (1) ≤ v (1) ≤ v (0) for t ∈ [0 , + ∞ ) and y ∈ [0 , l ]. By the method of induction, the sequences have thefollowing property ˆ v ≤ v ( m ) ≤ v ( m +1) ≤ v ( m +1) ≤ v ( m ) ≤ ˜ v for t ∈ [0 , + ∞ ) and y ∈ [0 , l ].Now we investigate the existence and uniqueness of a positive periodic solution of problem(3.5). Theorem 3.4 If R > , then problem (3 . admits a unique positive periodic solution v ∗ ( t, y ) . Proof:
First of all, we construct the upper solution of problem (3.5). Let ˜ v = M W ( t ) ( M > W ( t ) satisfying W t ( t ) = ( α − ˙ ρ ( t ) ρ ( t ) ) W ( t ) − γW ( t ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,W ( t ) = W ( t + T ) , t ≥ ,W (( nT ) + ) = g (cid:48) (0) W ( nT ) ≥ g ( W ( nT )) , n = 0 , , , ... (3.10)It is clear that ˜ v = M W ( t ) ( M >
1) is an upper solution of problem (3.5). Problem (3.10) canbe solved by direct calculations. Integrating from ( nT ) + to t ( t ∈ (( nT ) + , ( n + 1) T ]) in the firstequation in (3.10) yields W ( t ) = e αt W (( nT ) + ) W (( nT ) + ) (cid:82) tnT γeατρ ( τ ) dτ + e αnT , t ∈ (( nT ) + , ( n + 1) T ] , (3.11) hen W (( n + 1) T ) = e α ( n +1) T g (cid:48) (0) W ( nT ) g (cid:48) (0) W ( nT ) (cid:82) ( n +1) TnT γeατρ ( τ ) dτ + e αnT = e α ( n +1) T g (cid:48) (0) W ( nT ) g (cid:48) (0) W ( nT ) (cid:82) T γeα ( τ + nT ) ρ ( τ ) dτ + e αnT = e αT g (cid:48) (0) W ( nT ) g (cid:48) (0) W ( nT ) (cid:82) T γeατρ ( τ ) dτ +1 , Since R >
1, then αT > dλ (cid:82) T ρ ( t ) dt − ln g (cid:48) (0) > − ln g (cid:48) (0), that is, e αT g (cid:48) (0) >
1. Using theperiodicity, we obtain that W ( nT ) = e αT g (cid:48) (0) − g (cid:48) (0) (cid:82) T γeατρ ( τ ) dτ > W ( t ) = e αt ( e αT g (cid:48) (0) − e αT g (cid:48) (0) − (cid:82) tnT γe ατ ρ ( τ ) dτ + e αnT (cid:82) T γe ατ ρ ( τ ) dτ , t ∈ (( nT ) + , ( n + 1) T ] . So now we have the upper solution ˜ v of problem (3.5).Next we consider the lower solution and defineˆ v ( t, y ) = εφ ( nT, y ) , t = nT,ε ρ g (cid:48) (0) φ (( nT ) + , y ) , t = ( nT ) + ,ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ ( t, y ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ..., (3.12)where the positive eigenfunction φ ( t, y ) is defined in (2.4) and ε ( >
0) is a sufficiently small con-stant to be chosen later, as well as positive constants δ = α (1 − R ) and ρ = e − α (1 − R ) T g (cid:48) (0)are chosen to make sure that ˆ v ( nT, y ) = ˆ v (( n +1) T, y ). For t ∈ (( nT ) + , ( n +1) T ] and y ∈ (0 , l ),if ε < ε := δγ , we have ∂ ˆ v∂t − [ dρ ( t ) ˆ v + ( α − ˙ ρ ( t ) ρ ( t ) )ˆ v − γ ˆ v ]= [ α (1 − R ) − δ ] ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ + ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) [ dρ ( t ) φ yy + ( αR − ˙ ρ ( t ) ρ ( t ) ) φ ] − [ dρ ( t ) ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ yy + ( α − ˙ ρ ( t ) ρ ( t ) ) ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ ]+ γ ( ε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ ) = ε [ − δ + γε ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ ] ρ g (cid:48) (0) e [ α (1 − R ) − δ ]( t − nT ) φ< . On the other hand it follows from assumption (B2) that g (ˆ v ( nT, y )) − ˆ v (( nT ) + , y )= g (ˆ v ( nT, y )) − ε ρ g (cid:48) (0) φ (( nT ) + , y ) = g (ˆ v ( nT, y )) − ρ ˆ v ( nT, y ) ≥ ( g (cid:48) (0) − ρ )ˆ v ( nT, y ) − D { εφ ( nT, y ) } ν = [( g (cid:48) (0) − ρ ) − D ( εφ ( nT, y )) ν − ] εφ ( nT, y ) ≥ ε < ε := ( g (cid:48) (0) − ρ D ) ν − . These results show that ˆ v ( t, y ) is a lower solution of problem (3.5).Next, using v (0) = ˜ v and v (0) = ˆ v as the initial iteration, two sequences { v ( m ) } and { v ( m ) } are constructed from problem (3.7). By Lemma 3.2, we have thatˆ v ≤ v ( m ) ≤ v ( m +1) ≤ v ( m +1) ≤ v ( m ) ≤ ˜ v or every m = 1 , , ... Hence, the limits of sequences { v ( m ) } and { v ( m ) } exist andlim m →∞ v ( m ) = v ∗ , lim m →∞ v ( m ) = v ∗ , where v ∗ and v ∗ are T-periodic solutions of problem (3.5). Furthermore,ˆ v ≤ v ( m ) ≤ v ( m +1) ≤ v ∗ ≤ v ∗ ≤ v ( m +1) ≤ v ( m ) ≤ ˜ v. Now we claim that v ∗ and v ∗ are the maximal and minimal positive periodic solutions ofproblem (3.5). For any positive periodic solution v ( t, y ) of problem (3.5) satisfying ˆ v ≤ v ≤ ˜ v ,we use the same iteration as in (3.7) with the initial iteration v (0) = ˜ v and v (0) = v , where ˜ v and v are a pair of upper and lower solutions of problem (3.5), from which we can deduce that v ( t, y ) ≤ ¯ v ∗ ( t, y ) , t ≥ , y ∈ [0 , l ] , so that v ∗ is the maximal positive periodic solution of problem (3.5). Similarly, v ∗ is theminimal positive periodic solution of problem (3.5).Now the proof ends with showing the uniqueness of the positive periodic solution of problem(3.5). Let v and v be two solutions, and define sectorΛ = { s ∈ [0 , , sv ≤ v , t = 0 , t = 0 + , t ∈ (0 + , T ] , y ∈ [0 , l ] } It is easily seen that Λ contains a neighbourhood near by 0. We claim that 1 ∈ Λ. If not, weassume that s = sup Λ <
1. Recalling F ( v, t ) = f ( v, t ) + K ∗ v is nondecreasing and f ( v,t ) v isdecreasing in v on [0 , max v ], we deduce that( v − s v ) t − dρ ( t ) ( v − s v ) yy + K ∗ ( v − s v )= f ( v , t ) + K ∗ v − s ( f ( v , t ) + K ∗ v ) ≥ f ( s v , t ) + K ∗ s v − s ( f ( v , t ) + K ∗ v ) ≥ t ∈ (0 + , T ] and y ∈ (0 , l ). Using assumptions (A1) and (B1) we find that v (0 + , y ) − s v (0 + , y ) = g ( v (0 , y )) − s g ( v (0 , y )) ≥ g ( s v (0 , y )) − s g ( v (0 , y )) ≥ y ∈ (0 , l ). On the other hand, for t > v ( t, − s v ( t,
0) = v ( t, l ) − s v ( t, l ) = 0 . By the strong maximum principle [29], we have the following assertions:(i) v − s v > t = 0 + , t ∈ (0 + , T ] and y ∈ (0 , l ). Recalling v and v areT-periodic solutions, that is v (0 , y ) = v ( T, y ) and v (0 , y ) = v ( T, y ) for y ∈ (0 , l ), then v − s v > t ∈ [0 , T ] and y ∈ (0 , l ). By Hopf’s boundary lemma, ∂∂η | y =0 ( v − s v ) > ∂∂η | y = l ( v − s v ) < t ∈ [0 , T ], where η is the outward unit normalvector. Then there exists a constant (cid:15) > v − s v ≥ (cid:15)v , which leads to s + (cid:15) ∈ Λ.This contradicts the maximum property of s .(ii) v − s v ≡ t = 0 + , t ∈ (0 + , T ] and y ∈ (0 , l ). We then obtain that f ( v , t ) = s f ( v , t ). However, since s < f ( v , t ) = f ( s v , t ) > s f ( v , t ). Hence this caseis impossible.We conclude that problem (3.5) has a unique positive periodic solution v ∗ ( t, y ).We have now established the existence and uniqueness of a positive periodic solution ofproblem (3.5) in the above theorem. Now we show how this periodic solution is a globalattractor for the problem (1.11) This shown in the following theorem . heorem 3.5 If R > , then for nonnegative nontrivial initial value v ( y ) , any solution v ( t, y ) of problem (1 . satisfies lim m →∞ v ( t + mT, y ) → v ∗ ( t, y ) , t ≥ , y ∈ [0 , l ] , where v ∗ ( t, y ) is a positive T-periodic solution of problem (3 . . That is, v ∗ ( t, y ) is a globalattractor of problem (1 . . Proof:
Without loss of generality, we assume that v ( y ) > y ∈ (0 , l ). Otherwise we canreplace the initial time 0 by any t > v ( t , y ) > y ∈ (0 , l ). Noting that φ y (0 , > φ y (0 , l ) <
0, by Hopf’s boundary lemma, we can choose a sufficiently small ε such that εφ (0 , y ) ≤ v (0 , y ). Also, a sufficiently big M can be chosen such that v (0 , y ) ≤ M W (0). Forgiven ε and M , the function ˜ v := M W ( t ) with W ( t ) defined in (3.10) and ˆ v defined in (3.12),satisfies ˆ v (0 , y ) ≤ v (0 , y ) ≤ ˜ v (0 , y ) , y ∈ [0 , l ] . (3.13)Since g is nondecreasing with respect to v , we have thatˆ v (0 + , y ) = g (ˆ v (0 , y )) ≤ g ( v (0 , y )) = v (0 + , y ) ≤ g (˜ v (0 , y )) = ˜ v (0 + , y ) . It follows from the classical comparison principle that ˆ v ( t, y ) ≤ v ( t, y ) ≤ ˜ v ( t, y ) , t ∈ (0 + , T ] , y ∈ [0 , l ]. Induction shows that ˆ v ( t, y ) ≤ v ( t, y ) ≤ ˜ v ( t, y ) , t = nT, ( nT ) + , t ∈ (( nT ) + , ( n +1) T ] , y ∈ [0 , l ]. Hence, for n = 0 , , , ... , v (0) ( t, y ) ≤ v ( t, y ) ≤ v (0) ( t, y ) , t = nT, ( nT ) + , t ∈ (( nT ) + , ( n + 1) T ] , y ∈ [0 , l ] . (3.14)Then, v (0) ( T, y ) ≤ v ( T, y ) ≤ v (0) ( T, y ) , y ∈ [0 , l ] , (3.15)which together with v (1) (0 , y ) = v (0) ( T, y ) and v (1) (0 , y ) = v (0) ( T, y ) yields v (1) (0 , y ) ≤ v ( T, y ) ≤ v (1) (0 , y ) , y ∈ [0 , l ] . By the monotonicity of g and (3.15), we obtain g ( v (0) ( T, y )) ≤ g ( v ( T, y )) ≤ g ( v (0) ( T, y )) , y ∈ [0 , l ] . Due to the last two equations in (3.7) and the last equation in (1.11) we have that v (1) (0 + , y ) = g ( v (0) ( T, y )) ≤ g ( v ( T, y )) = v ( T + , y ) ≤ g ( v (0) ( T, y )) = v (1) (0 + , y ) , y ∈ [0 , l ] , that is, v (1) (0 + , y ) ≤ v ( T + , y ) ≤ v (1) (0 + , y ) , y ∈ [0 , l ] . Then, from the classical comparison principle v (1) ( t, y ) ≤ v ( t + T, y ) ≤ v (1) ( t, y ) holds for t ∈ (0 + , T ] and y ∈ [0 , l ]. By induction we have v (1) ( t, y ) ≤ v ( t + T, y ) ≤ v (1) ( t, y ) , t = nT, ( nT ) + , t ∈ (( nT ) + , ( n + 1) T ] , y ∈ [0 , l ]for n = 0 , , , ... Also by induction, it follows from the last two equations in problem (3.7) that v ( m ) ( t, y ) ≤ v ( t + mT, y ) ≤ v ( m ) ( t, y ) , t ≥ , y ∈ [0 , l ] , (3.16)since (3.16) holds for m = 0 and m = 1. With lim m →∞ v ( m ) ( t, y ) = lim m →∞ v ( m ) ( t, y ) = v ∗ ( t, y ) bythe uniqueness of the periodic solution of problem (3.5) presented in Theorem 3.4, we concludethat lim m →∞ v ( t + mT, y ) → v ∗ ( t, y ) , t ≥ , y ∈ [0 , l ] . .2 Nonmonotone Case To investigate the nonmonotone case we need the following restriction, just like in [17]:(B3) There exists a σ > g ( u ) is nondecreasing for 0 ≤ u ≤ σ .This assumption is consistent with the actual harvesting mechanism. The classical plusefunction, e.g the Ricker function g ( u ) = ue r − bu , satisfies this; g ( u ) is increasing for 0 < u < /b and decreasing for u > /b .We define g + ( u ) = max ≤ w ≤ u g ( w ) (3.17)for u ≥
0. Obviously, for u ≥ g + ( u ) is nondecreasing and g + ( u ) ≥ g ( u ), g + ( u ) = g ( u ) forsmall u > g + (cid:48) (0) = g (cid:48) (0). Under the condition R >
1, the following problem admits aminimal positive periodic solution v + ( t ) satisfying v t ( t ) = ( α − ˙ ρ ( t ) ρ ( t ) ) v ( t ) − γv ( t ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...,v (0) = v ( T ) , t ≥ ,v (( nT ) + ) = g + ( v ( nT )) , n = 0 , , , ... (3.18)Let β + = min t ∈ [0 ,T ] v + ( t ). We also define g − ( u ) = min u ≤ w ≤ β + g ( w ) (3.19)for 0 ≤ u ≤ β + . It is obvious to see that for u ≥ g − ( u ) is nondecreasing and g − ( u ) ≤ g ( u ), g − ( u ) = g ( u ) for small u > g −(cid:48) (0) = g (cid:48) (0).Then we take the following two auxiliary problems into account v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v − γv , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = v ( y ) ≥ , (cid:54)≡ , y ∈ [0 , l ] ,v (( nT ) + , y ) = g + ( v + ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ..., (3.20)and v t = dρ ( t ) v yy + ( α − ˙ ρ ( t ) ρ ( t ) ) v − γv , y ∈ (0 , l ) , t ∈ (( nT ) + , ( n + 1) T ] , n = 0 , , , ...v ( t,
0) = v ( t, l ) = 0 , t > ,v (0 , y ) = v ( y ) ≥ , (cid:54)≡ , y ∈ [0 , l ] ,v (( nT ) + , y ) = g − ( v − ( nT, y )) , y ∈ (0 , l ) , n = 0 , , , ..., (3.21)We use v + ( t, y ) and v − ( t, y ) to represent the solution of problem (3.20) and (3.21), respectively.Recalling that g − ( v ) ≤ g ( v ) ≤ g + ( v ) for u ≥
0, Lemma 3.1 implies that if 0 ≤ v +0 ( y ) ≤ v ( y ) ≤ v − ( y ) ≤ β + , then any solution v ( t, y ) of problem (1.11) satisfies0 ≤ v − ( t, y ) ≤ v ( t, y ) ≤ v + ( t, y ) ≤ β + . (3.22) y expression (2.5), problem (3.20) and (3.21) have the same basic reproduction number R .By Theorems 3.2, 3.4 and 3.5, problem (3.20) and (3.21) have the same asymptotic behaviorsof solutions. Owing to this and (3.22), when R <
1, we obtain that the solution v ( t, y ) ofproblem (1 .
11) converges to 0 as t → ∞ . If R >
1, then the solution v − ( t, y ) of problem (3.21)satisfies v − ( t, y ) ≤ lim inf m → + ∞ v − ( t + mT, y ) ≤ lim sup m → + ∞ v − ( t + mT, y ) ≤ v − ( t, y ) , t ≥ , y ∈ [0 , l ] , where v − ( t, y ) and v − ( t, y ) are the minimum and maximum positive periodic solutions of thecorresponding periodic problem related to problem (3.21), respectively. By (3.22), we candeduce that any positive solution v ( t, y ) of problem (1.11) satisfieslim inf m → + ∞ v ( t + mT, y ) ≥ v − ( t, y ) . Hence, the asymptotic behavior of the solution to problem (1.11) can be given as follows:
Theorem 3.6 ( i ) If R < , then the solution v ( t, y ) of problem (1 . satisfies lim t →∞ v ( t, y ) = 0 . ( ii ) If R > , then any solution v ( t, y ) of problem (1 . satisfies lim inf m → + ∞ v ( t + mT, y ) ≥ v − ( t, y ) provided that (cid:54)≡ , ≤ v ( y ) ≤ β + , where v − ( t, y ) is the minimal positive periodic solution of thecorresponding periodic problem related to problem (3.21). The above sections show the threshold-type dynamics for a population, determined by diffusionparameters, evolution rate and properties of the impulsive harvesting. In this section, we aimto investigate how the evolution rate and impulsive harvesting affect these dynamical behaviorsof a population by adopting numerical analyses. In all simulations, we consider the interval[0 , l ( t )] = [0 , ρ ( t ) l ], where l = π , and we fix some parameters d = 1 , α = 1 . , γ = 0 . λ = ( πl ) = 1. We choose v ( y ) = 0 . x ) + 0 . x ) as initialfunction. We choose two different ρ ( t ) to analyze the effect of the evolution rate on the dynamics of thepopulation when the impulsive harvesting occurs every time T = 2. We first fix g ( u ) as theBeverton-Holt function with a = 10 and m = 8 as in (1.2), in which case g ( u ) is a monotonicallyincreasing function. xample 4.1 Fix g ( u ) = 8 u/ (10+ u ) and then g (cid:48) (0) = 0 . . We first choose ρ ( t ) = e − . − cos πt ) ,it follows from (2.5) that R ( ρ ) = α dλ T (cid:82) T ρ ( t ) dt − T ln g (cid:48) (0) ≈ . < . One can see from Fig. 1 that the population suffers eventual extinction.Now we choose ρ ( t ) = e . − cos πt ) . Direct calculations show that R ( ρ ) = α dλ T (cid:82) T ρ ( t ) dt − T ln g (cid:48) (0) ≈ . > . It is easily seen from Fig. 2 that the population approaches a positive periodic steady state.The example shows that the population vanishes in a periodically evolving habitat with asmall evolution rate and persists in a habitat with a larger evolution rate. (a) (b)(c)
Figure 1: ρ ( t ) = e − . − cos πt ) and g ( u ) equals the Beverton-Holt function with a = 10 and m = 8. The domain isperiodically evolving with ρ and R <
1. Graphs ( a ) − ( c ) show that the population u ( t, x ) decays to 0. Graphs ( b ) and( c ) are the cross-sectional view and projection of u on the t − u -plane, respectively. The color bar in graph ( b ) shows thedensity of u ( t, x ). Now we consider the case when g is nonmonotone. We fix g ( u ) as the Ricker function with b = 1 . r = 0 .
05 as in (1.3), then g ( u ) satisfies assumptions (A1) and (B3). Example 4.2
We fix g ( u ) = ue . − . u , then g (cid:48) (0) = e . . As Example 4.1, we choose ρ ( t ) = e − . − cos πt ) firstly, then R ( ρ ) = α dλ T (cid:82) T ρ ( t ) dt − T ln g (cid:48) (0) ≈ . < . a) (b)(c) Figure 2: ρ ( t ) = e . − cos πt ) and g ( u ) equals the Beverton-Holt function with a = 10 and m = 8, then R > u ( t, x ) are shown in graph ( a ), indicating that the population approaches apositive periodic steady state. Graph ( b ) is the cross-sectional view and indicates the periodic evolution of the domain.The appearance of impulsive harvesting every time T = 2 can be seen in graph ( c ), which is the projection of u on the t − u -plane. One can see from Fig. 3 that the population eventually suffers extinction.We next choose ρ ( t ) = e . − cos πt ) . Direct calculations show that R ( ρ ) = α dλ T (cid:82) T ρ ( t ) dt − T ln g (cid:48) (0) ≈ . > . It is easily seen from Fig. 4 that the population approaches a positive periodic steady state.It can be seen from Example 4.2 that when impulsive harvesting occurs in the form ofthe Ricker function, the population presents dynamics similar to the case where the impulsiveharvesting occurs in the form of the Beverton-Holt function, that is, the population vanishes ina periodically evolving habitat with a small evolution rate and persists in a habitat with a largerevolution rate.
It is shown in Examples 4.1 and 4.2 that the evolution rate of the domain can imposesimilar dynamical behaviors of the population, no matter if g ( u ) is given as the Beverton-Holtfunction or the Ricker function. We note that the evolution rate of domain plays an importantrole in persistence and extinction of the population, that is, the larger the evolution rate is, themore beneficial it is for the populations survival. We also note that a large domain evolutionrate has a positive effect on the survival of the population when an impulsive harvesting takesplace. a) (b)(c) Figure 3: ρ ( t ) = e − . − cos πt ) and g ( u ) equals the Ricker function with b = 1 . r = 0 .
05. The domain isperiodically evolving with ρ and R <
1. Graphs ( a ) − ( c ) show that the population u ( t, x ) decays to 0. In order to investigate how impulsive harvesting affects the dynamics of the population in aperiodically evolving habitat, we employ numerical simulations to compare situations whenimpulsive harvesting does occur or not. We firstly consider the case when monotone impulsiveharvesting takes place, and g ( u ) is chosen as the Beverton-Holt function. Example 4.3
We first fix ρ ( t ) = e − . − cos πt ) . It follows from (2.9) that R ( ρ ) ≈ . < and impulsive harvesting does not occur. One can see from Fig. 5 that the population suffersextinction eventually. Comparing Fig. 1 to Fig. 5, we can conclude that when impulsiveharvesting occurs, population suffers extinction at a faster speed.Now we fix ρ ( t ) = e . − cos πt ) . Now, without impulsive harvesting, we have R ( ρ ) ≈ . > . It is easily seen from Fig. 6 that the population stabilizes to a positive periodicsteady state. Then we choose g ( u ) = 5 u/ (10 + m ) . One can see from Fig. 7 that the populationnow decays to extinction. We note from Figs. 6 and 7 that the population survives in a evolvingdomain with a large evolution rate, but vanishes when the impulsive harvesting takes place. Now we discuss the cases where nonmonotone impulsive harvesting takes place, and g ( u )is chosen as the Ricker function. Example 4.4
Fix ρ ( t ) = e − . − cos πt ) . A comparison of Figs. 3 with impulses and 5 withoutimpulses reveals that impulsive harvesting accelerates the extinction of the population.We next fix ρ ( t ) = e . − cos πt ) and choose g ( u ) = ue . − u . Fig. 8 shows that the popu-lation approaches a very small positive steady state. Taken together, Figs. 6 without impulsesand 8 without impulses indicate that the population develops well in an evolving domain with a) (b)(c) Figure 4: ρ ( t ) = e . − cos πt ) and g ( u ) equals the Ricker function with b = 1 . r = 0 .
05. In this situation, R >
1. Graphs ( a ) − ( c ) indicate that the population approaches a positive periodic steady state. a large evolution rate, and when impulsive harvesting occurs, the size of the population dropssharply at the beginning but eventually survives. Examples 4.3 and 4.4 indicate that when the population lives in a periodically evolvinghabitat with a small evolution rate, impulsive harvesting can speed up the extinction of thepopulation no matter which kind of impulse function is chosen. We see that impulsive har-vesting has a negative effect on the survival of the population and can even can lead to theextinction of the population.
A diffusive logistic population model with impulsive harvesting on a periodically evolving do-main has been investigated in the present paper. What we want to study is how the evolutionrate of the domain and impulsive harvesting affect on the dynamics of the population. Toaddress this, we firstly introduce a new threshold value R for this impulsive problem; this isthe ecological reproduction index, and is given by an explicit formula and it mainly dependson the evolution rate ρ ( t ) and the term g (cid:48) (0), where g is the pulse function modelling the im-pulsive harvesting. Then, considering two cases where the function g ( u ) is either monotone ornot, threshold-type dynamical behaviors of the solution to problem (1.11) are established. Weconclude that when R is smaller than 1, the solution v ( t, y ) decays to 0 as time goes to infinityno matter which function g ( u ) is chosen (see Theorems 3.2 and 3.7(i)). On the other hand,when R is greater than 1, it is proved in Theorem 3.6 that the solution v ( t, y ) converges to apositive periodic steady state in the monotone case, while in the nonmonotone case, solution v ( t, y ) converges to an attractive sector (see Theorem 3.7(ii)). a) (b)(c) Figure 5:
A numerical simulation of the population without impulses and ρ ( t ) = e − . − cos πt ) . In this case, R < a ) − ( c ) show that the population u ( t, x ) decays to 0. Graphs ( b ) and ( c ) are the cross-sectional view and projectionof u on the plane t − u , respectively. Our numerical simulations further reveal the effects of the evolution rate of the domain andimpulsive harvesting on the persistence and extinction of the population. Examples 4.1 and 4.2illustrate how the population suffers extinction in a evolving habitat with a small evolution rate,but survives in one with a large evolution rate. That is, a large evolution rate can be beneficialfor the survival of the population no matter which kind of impulsive harvesting that occur.Another notable result is that impulsive harvesting can speed up the population extinction (seeFigs. 1, 3 and 5) and has negative effect on the population survival (see Figs. 6 and 8), andcan even lead to the extinction of the population (see Fig. 7).Our analyses and simulations are based on the one-dimensional case. Recently, Fazly, Lewisand Wang [13] have extended the results in [17] to a higher dimensional habitat without evolv-ing. They provided critical domain parameters and investigated the extinction and persistenceof species, depending on the geometry and size of the domain. We are curious about what neweffects the evolution rate will have on the population dynamics when a higher dimensional andevolving region is introduced. Due to the spatial heterogeneity, model parameters, which de-pend on location, can be considered. Furthermore, a hybrid reaction-advection-diffusion modelwith a nonlocal discrete time map have been very recently investigated by Fazly, Lewis andWang in [12]. They obtained the existence of travelling wave solutions and provided explicitexpressions for the spreading speed. If the nonlocal discrete time map is introduced to ourmodel, it will be interesting to investigate how this will affect the dynamics of the population.This challenges us to a further study. a) (b)(c) Figure 6:
A numerical simulation of the population without impulses and ρ ( t ) = e . − cos πt ) . In this situation, R >
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